Shape Loop Space of Pro-discrete Spaces

Document Type : Research Articles

Author

Department of Pure Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

Abstract

In this paper, considering the $k$th shape loop space $\check{\Omega}_{k}^{\mathbf{p}}(X,x)$, for an HPol$_*$-expansion $\mathbf{p}:(X,x)\rightarrow ((X_{\lambda},x_{\lambda}),[p_{\lambda\lambda'}],\Lambda)$ of a pointed topological space $(X,x)$, first we prove that  $\check{\Omega}_{k}$ commutes with the product under some conditions and then  we show that $\check{\Omega}_k^{\mathbf{p}}(X,x)\cong \displaystyle{\lim_{\leftarrow}\check{\Omega}_k^{\mathbf{p}}(X_i,x_i)}$, for a pro-discrete space $(X,x)=\displaystyle{\lim_{\leftarrow}(X_i,x_i)}$ of compact polyhedra. Finally,  we conclude that these spaces are metric, second countable and separable.

Keywords

Caspian Journal of Mathematical Sciences
Volume 11, Issue 1
2022
Pages 242-249

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